140 research outputs found
Complete intersections in binomial and lattice ideals
For the family of graded lattice ideals of dimension 1, we establish a
complete intersection criterion in algebraic and geometric terms. In positive
characteristic, it is shown that all ideals of this family are binomial set
theoretic complete intersections. In characteristic zero, we show that an
arbitrary lattice ideal which is a binomial set theoretic complete intersection
is a complete intersection.Comment: Internat. J. Algebra Comput., to appea
Prime splittings of Determinantal Ideals
We consider determinantal ideals, where the generating minors are encoded in
a hypergraph. We study when the generating minors form a Gr\"obner basis. In
this case, the ideal is radical, and we can describe algebraic and numerical
invariants of these ideals in terms of combinatorial data of their hypergraphs,
such as the clique decomposition. In particular, we can construct a minimal
free resolution as a tensor product of the minimal free resolution of their
cliques. For several classes of hypergraphs we find a combinatorial description
of the minimal primes in terms of a prime splitting. That is, we write the
determinantal ideal as a sum of smaller determinantal ideals such that each
minimal prime is a sum of minimal primes of the summands.Comment: Final version to appear in Communications in Algebr
Colorings and Sudoku Puzzles
Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\ obner bases in multivariate polynomial rings. We utilize Gr\ obner bases of an ideal of a multivariate polynomial ring over a finite field to solve coloring, Shidoku, and Sudoku problems. In the last section, we note Gr\ obner bases are also well-suited to hypergraph coloring problems
An Abstraction of Whitney's Broken Circuit Theorem
We establish a broad generalization of Whitney's broken circuit theorem on
the chromatic polynomial of a graph to sums of type
where is a finite set and is a mapping from the power set of into
an abelian group. We give applications to the domination polynomial and the
subgraph component polynomial of a graph, the chromatic polynomial of a
hypergraph, the characteristic polynomial and Crapo's beta invariant of a
matroid, and the principle of inclusion-exclusion. Thus, we discover several
known and new results in a concise and unified way. As further applications of
our main result, we derive a new generalization of the maximums-minimums
identity and of a theorem due to Blass and Sagan on the M\"obius function of a
finite lattice, which generalizes Rota's crosscut theorem. For the classical
M\"obius function, both Euler's totient function and its Dirichlet inverse, and
the reciprocal of the Riemann zeta function we obtain new expansions involving
the greatest common divisor resp. least common multiple. We finally establish
an even broader generalization of Whitney's broken circuit theorem in the
context of convex geometries (antimatroids).Comment: 18 page
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